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Constraints on scalar spectral index from latest observational measurements
Physics of the Dark Universe 2 (2013) 188–194
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Physics of the Dark Universe journal homepage: www.elsevier.com/locate/dark
Constraints on scalar spectral index from latest observational measurements夽 Hong Lia , b , * , Jun-Qing Xiaa , Xinmin Zhangc a
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Science, P.O. Box 918-3, Beijing 100049, PR China National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, PR China c Theoretical Physics Division, Institute of High Energy Physics, Chinese Academy of Science, P.O. Box 918-4, Beijing 100049, PR China b
a r t i c l e
i n f o
Keywords: CMB The scalar spectral index
a b s t r a c t With the nine-year data release of the Wilkinson Microwave Anisotropy Probe (WMAP9), it is found that the inflationary models with the scalar spectral index ns ≥ 1 are excluded at about 5σ confidence level. In this paper, we set the new limits on the scalar spectral index in different cosmological models by the WMAP9 data, the small-scale cosmic microwave background (CMB) measurement from the South Pole Telescope, baryon acoustic oscillation data, Hubble Telescope measurements of the Hubble constant, and supernovae luminosity distance data. In most of extended cosmological models, e.g. with a dark energy equation of state, the constraints on ns do not change significantly, when comparing with that obtained in the standard CDM model. The Harrison–Zel’dovich–Peebles (HZ) scale invariant spectrum is still disfavored at more than 4σ confidence level. However, when considering the model with an effective number of neutrinos Neff , we obtain the limit on the spectral index of ns = 0.980 ± 0.011 (1σ ), due to the strong degeneracy between ns and Neff . The HZ spectrum now is consistent with the current data at 95% confidence level. Recently, the Planck collaboration has published CMB maps with the highest precision. Therefore, we also analyze these extended cosmological models again using the Planck data, and find that the degeneracy between ns and Neff still weakens the constraint on the spectral index significantly. c 2013 The Authors. Published by Elsevier B.V. All rights reserved.
1. Introduction Inflation, the most attractive paradigm in the very early universe, has successfully resolved many problems existing in the hot big bang cosmology, such as the flatness, horizon, monopole problem, and so forth [1]. Its quantum fluctuations turn out to be the primordial density fluctuations which seed the observed large scale structures (LSS) and the anisotropies of cosmic microwave background (CMB). To distinguish various inflationary models, the spectral index of the power spectrum of primordial curvature perturbations is one of the most important variables. With the accumulation of observational data from CMB, LSS and Type Ia Supernovae observations (SN) and the improvements of the data quality, the cosmological observations play a crucial role in our understanding of the Universe and also in constraining the cosmological parameters [2–5]. Thus, determining the scalar spectral index
夽 This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike License, which permits non-commercial use, distribution, and reproduction in any medium, provided the original author and source are credited. * Corresponding author at: Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Science, P.O. Box 918-3, Beijing 100049, PR China. Tel.: +86 10 88235124. E-mail address: [email protected] (H. Li).
ns from the observational data is a very powerful and reliable way to understand these inflationary models. The nine-year data release of Wilkinson Microwave Anisotropy Probe (WMAP9) has determined the cosmological parameters accurately and found that the 68% C.L. constraint on the scalar spectral index of ns = 0.9608 ± 0.0080 [6], when combining with the smallscale CMB measurement from the South Pole Telescope (SPT), baryon acoustic oscillation data (BAO) and Hubble Telescope measurements of the Hubble constant (HST). Within the CDM framework, the Harrison–Zel’dovich–Peebles (HZ) scale invariant spectrum (ns ≡ 1) and the spectra with ns > 1 are disfavored at about 5σ confidence level. Recently, the Planck collaboration has released the first cosmological papers providing the highest resolution, full sky, maps of the CMB temperature anisotropies [7]. Due to the improved precision, this new Planck data has constrained several cosmological parameters at few percent level. In the CDM model, the constraint on the spectral index is significantly improved by the new Planck data, namely ns = 0.9603 ± 0.0054 (1σ ) [8]. The spectra with ns ≥ 1 are ruled out at about 8σ confidence level. Although the CDM model is a good candidate for interpreting the data [8], it is still very interesting to investigate constraints on the scalar spectral index in various extended cosmological models, such as the effect number of neutrinos [9,10], the fraction of baryonic mass in primordial helium, the massive neutrino [11,12] or the
c 2013 The Authors. Published by Elsevier B.V. All rights reserved. 2212-6864/$ - see front matter http://dx.doi.org/10.1016/j.dark.2013.11.003
H. Li et al. / Physics of the Dark Universe 2 (2013) 188–194
equation of state of dark energy [13]. More importantly, the degeneracies between ns and these cosmological parameters introduced could weaken the constraints on ns [8,14–18]. In this paper, we explore the cosmological constraints on ns in some extended cosmological models from the latest data sets, including the Planck and WMAP9 power spectra, the small-scale CMB measurement from SPT, the BAO measurements from several LSS surveys, the HST prior on the Hubble constant H0 and the “Union2.1” compilation SN sample made by the Supernova Cosmology Project. Firstly, we consider the general inflationary model with the tensor fluctuations (r) in the CDM framework. We then extend the CDM model allowing for the dark energy models with a constant equation of state (EoS, w) or with a time-varying EoS (w (z)). Finally, we include the massive neutrino case ( mν ) or the effective number of neutrinos (Neff ) into the CDM model. Our paper is organized as follows: In Section 2 we de scribe the method and the latest observational data sets used in the numerical analyses; Section 3 contains our main global constraints of the scalar spectral index ns in different cosmological models from the current observations. The last Section 4 is the conclusions. 2. Method and data 2.1. Numerical method We perform a global fitting of cosmological parameters using the CosmoMC package [19], a Markov Chain Monte Carlo code. We assume purely adiabatic initial conditions and a flat CDM Universe. The following six cosmological parameters are allowed to vary with top-hat priors: the cold dark matter energy density parameter c h2 ∈ [0.01, 0.99], the baryon energy density parameter b h2 ∈ [0.005, 0.1], the scalar spectral index ns ∈ [0.5, 1.5], the primordial amplitude ln [1010 As ] ∈ [2.7, 4.0], the ratio (multiplied by 100) of the sound horizon at decoupling to the angular diameter distance to the last scattering surface 100s ∈ [0.5, 10], and the optical depth to reionization τ ∈ [0.01, 0.8]. The pivot scale is set at ks 0 = 0.05 Mpc−1 . Besides these six basic cosmological parameters, we have several extra cosmological parameters in different extended cosmological models: the running of scalar spectral index α s ≡ d ln ns /d ln k ∈ [−0.1, 0.1]; the tensor to scalar ratio of the primordial spectrum r ≡ At /As ∈ [0, 2]; the fraction of baryonic mass in primordial helium Yp ∈ [0.1, 0.5], the total neutrino mass fraction at the present day
fν ≡
ν h2 mν = ∈ [0, 0.1] ; mh2 93.14 eV mh2
(1)
and the effective number of neutrinos Neff ∈ [0, 10]. We also consider the dark energy model with the EoS parameters w0 ∈ [−2, 0] and w1 ∈ [−5, 2], which is given by the parametrization [20] wde (a) = w0 + w1 (1 − a) ,
(2)
where a ≡ 1/(1 + z) is the scale factor and w1 = −dw/da characterizes the “running” of EoS. The CDM model has w0 = −1 and w1 = 0. For the dark energy model with a constant EoS, w1 = 0. When using the global fitting strategy to constrain the cosmological parameters, it is crucial to include dark energy perturbations [21]. In this paper we use the method provided in Refs. [21,22] to treat the dark energy perturbations consistently in the whole parameter space in the numerical calculations. Therefore, the most general parameter space in the analyses is: b h2 , c h2 , s , τ , ns , As , αs , r, w0 , w1 , Y p , fν , Nef f . (3)
2.2. Current observational data In our analysis, we consider the following cosmological probes: (i) power spectra of CMB temperature and polarization anisotropies;
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(ii) the baryon acoustic oscillation in the galaxy power spectra; (iii) measurement of the current Hubble constant; and (iv) luminosity distances of type Ia supernovae. To incorporate the WMAP9 CMB temperature and polarization power spectra, we use the routines for computing the likelihood supplied by the WMAP team [6]. The WMAP9 polarization data are composed of TE/EE/BB power spectra on large scales (2 ≤ ≤ 23) and TE power spectra on small scales (24 ≤ ≤ 800), while the WMAP9 temperature data includes the CMB anisotropies on scales 2 ≤ ≤ 1200. Furthermore, we also use the recent SPT data [23], using 47 bandpowers in the range 600 ≤ ≤ 3000. The likelihood is assumed to be Gaussian, and we use the published bandpower window functions and covariance matrix. In order to address for foreground contributions, the SZ amplitude, the amplitude of the clustered point source contribution, and the amplitude of the Poisson distributed point source contribution, are added as nuisance parameters in the CMB data analyses. For the Planck data from the 1-year data release [7], we use the low- and high- CMB temperature power spectrum data from Planck with the low- WMAP9 polarization data (Planck + WP). We marginalize over the nuisance parameters that model the unresolved foregrounds with wide priors [24], and do not include the CMB lensing data from Planck [25]. Baryon Acoustic Oscillations provides an efficient method for measuring the expansion history by using features in the clustering of galaxies within large scale surveys as a ruler with which to measure the distance–redshift relation. It provides a particularly robust quantity to measure [26]. It measures not only the angular diameter distance, DA (z), but also the expansion rate of the universe, H(z), which is powerful for studying dark energy [27]. Since the current BAO data are not accurate enough for extracting the information of DA (z) and H(z) separately [28], one can only determine an effective distance [29]: 1/3 D ν (z) = (1 + z)2 D 2A (z) cz/ H (z) .
(4)
In this paper we use the recent BAO measurement at high redshift z = 2.3 detected in the Ly-α forest of Baryon Oscillation Spectroscopic Survey (BOSS) quasars [30]. Furthermore, we also include the BAO measurement from the 6dF Galaxy Redshift Survey (6dFGRS) at a low redshift z = 0.106 [31], and the BAO measurements from the WiggleZ Survey at three redshift bins z = 0.44, z = 0.60 and z = 0.73 [32], the measurement of the BAO scale based on a re-analysis of the Luminous Red Galaxies (LRG) sample from Sloan Digital Sky Survey (SDSS) Data Release 7 at the median redshift z = 0.35 [33], and the BAO signal from BOSS CMASS DR9 data at z = 0.57 [34]. In our analysis, we add a Gaussian prior on the current Hubble constant given by Ref. [35]; H0 = 73.8 ± 2.4 km s−1 Mpc−1 (68% C.L.). The quoted error includes both statistical and systematic errors. This measurement of H0 is obtained from the magnitude–redshift relation of 240 low-z Type Ia supernovae at z < 0.1 by the Near Infrared Camera and Multi-Object Spectrometer (NICMOS) Camera 2 of the Hubble Space Telescope (HST). In addition, we impose a weak top-hat prior on the Hubble parameter: H0 ∈ [40, 100] km s−1 Mpc−1 . Finally, we include data from Type Ia supernovae, which consists of luminosity distance measurements as a function of redshift, DL (z). In this paper we use the latest SN data sets from the Supernova Cosmology Project, “Union Compilation 2.1”, which consists of 580 samples and spans the redshift range 0 ≤ z ≤ 1.55 [36]. This data set also provides the covariance matrix of data with and without systematic errors. In order to be conservative, we use the covariance matrix with systematic errors. When calculating the likelihood from SN, we marginalize over the absolute magnitude M, which is a nuisance parameter, as done in Ref. [37].
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Table 1 1σ constraints on some cosmological parameters from different data combinations in the standard CDM model. Standard CDM WMAP9 alone WMAP9 + SPT ns H0 100 b h2 100 c h2
0.972 70.34 2.270 11.37
± ± ± ±
0.013 2.21 0.050 48
0.966 70.20 2.234 11.41
± ± ± ±
0.011 2.15 0.042 0.47
All datasets 0.963 68.90 2.224 11.70
± ± ± ±
0.008 0.62 0.034 0.17
3. Numerical results In this section we mainly present our global fitting results of the cosmological parameters determined from the latest observational data and focus on the degeneracies between ns and other extended parameters in different models. And finally, we report the constraints on the scalar spectral index from the recent Planck data. 3.1. Standard ΛCDM model Firstly, we consider the standard CDM model. In Table 1 we show the constraints on some related cosmological parameters from three different data combinations: WMAP9 alone, WMAP9 + SPT, and all datasets. In the upper panel of Fig. 1 we show the one-dimensional marginalized likelihood distributions of ns from three data combinations. Using the WMAP9 data alone, we obtain the 68% constraint of ns = 0.972 ± 0.013. The primordial spectra with ns ≥ 1 are only excluded at 2σ confidence level. When we include the smallscale SPT measurement, the constraint on ns becomes slightly tighter, ns = 0.966 ± 0.011 at 1σ confidence level. Since the median value and the error bar of ns are smaller, when comparing with those from WMAP9 alone, the significance of ns < 1 is more than 3σ confidence level. We also show the two-dimensional contour between H0 and ns in the below panel of Fig. 1. As we know, the Hubble constant is anticorrelated with the matter density m . Changing the matter density has effects on the small-scale power spectrum and it needs the spectral index ns to be changed to compensate. Therefore, there is a strong correlation between ns and H0 . When using all datasets together, this degeneracy can be partly broken by the information of H0 measurement. Thus, the constraint on the Hubble constant becomes much more stringent, H0 = 68.90 ± 0.62 km s−1 Mpc−1 (1σ C.L.), due to the HST prior. Consequently, the constraint of the spectral index also becomes tighter significantly, ns = 0.963 ± 0.008 (1σ C.L.). The error bar of ns is reduced by a factor of 1.5, due to the constraining power of BAO, HST and SN. The HZ spectrum is disfavored by the current data at about 5σ confidence level, which is consistent with that from the WMAP9 paper [6]. However, this strong constraint on the spectral index is model dependent apparently. The constraints on ns could be changed, due to the possible degeneracies between ns and other extended parameters in some extended CDM models. In the following subsections, we discuss the constraints on parameters in these extended cosmological models, which is shown in Table 2, such as the inflationary models, α s and r, the dynamical dark energy model, w(z), and ones including the neutrino properties, mν and Neff . 3.2. Inflationary models Firstly, we include the gravitational waves into the analysis. When using all datasets together, the data yield the 95% upper limit of tensor-to-scalar ratio r < 0.15. Meanwhile, the constraint on the spectral index is barely changed, ns = 0.966 ± 0.009 at 68% confidence level, due to the degeneracy between ns and r. In Fig. 2 we show the
Fig. 1. Marginalized one-dimensional and two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and H0 in the standard CDM model from different present data combinations: WMAP9 only (red), WMAP9 + SPT (blue) and all datasets (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 2 1σ constraints on cosmological parameters r, α s , mν , Yp and dark energy EoS from the data combination of WMAP9 + SPT + BAO + SN in different extended models. For the weakly constrained parameters we quote the 95% upper limits instead. Models
Constraints
ns Constraints
CDM + r CDM + α s WCDM W(z)CDM
r < 0.15 α s = −0.023 ± 0.011 w = −1.060 ± 0.066 w0 = −1.11 ± 0.15 w1 = 0.18 ± 0.65 mν < 0.47 eV Yp = 0.298 ± 0.031
0.966 0.948 0.960 0.962 – 0.968 0.976
CDM + mν CDM + Yp
± ± ± ±
0.009 0.011 0.009 0.011
± 0.009 ± 0.012
two-dimensional constraints in the (ns , r) panel which can be compared with the prediction of the inflation models. We find that the HZ scale-invariant spectrum (ns = 1, r = 0) is still disfavored at about 4σ confidence level. Also, the inflation models with “blue” tilt (ns > 1) are excluded by the current observations. Furthermore, assuming the number of e-foldings N = 50−60, the single slow-rolling scalar field with potential V(φ ) ∼ m2 φ 2 , which predicts (ns ,r) = (1 − 2/N, 8/N), is still within the 2σ region, while another single slowrolling scalar field with potential V(φ ) ∼λφ 4 , which predicts (ns ,r) = (1−3/N,16/N), has been excluded more than at 2σ confidence level. We also explore the constraint on the running of the spectral index from the latest observational data. When neglecting the tensor fluctuations (r = 0), the combination of the current observational data yield
H. Li et al. / Physics of the Dark Universe 2 (2013) 188–194
Fig. 2. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and r from all datasets together (red). The two blue solid lines are predicted by the m2 φ 2 and λφ 4 models, respectively. The green points denote predictions assuming that the number of e-foldings N = 50−60 from two models. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and α s from all datasets together with (blue) and without (red) considering the tensor fluctuations in the analysis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the limit on the running of the spectral index of α s = −0.023 ± 0.011 (1σ ), which means the running of ns is favored by the current data at 2σ confidence level. In Fig. 3 we show the two-dimensional constraints in the (ns ,α s ) panel. Due to the degeneracy between ns and α s , the 68% constraint on ns is slightly enlarged, ns = 0.948 ± 0.011. The error bar is relaxed by a factor of 1.5, when comparing with the standard CDM model. Finally, we vary the α s and r simultaneously in the analysis. From the blue contour of Fig. 3, one can see that the constraint on ns does not change, ns = 0.949 ± 0.011 (1σ ). The degeneracy between α s and r significantly weakens the constraints on them, namely the 68% constraint on α s is α s = −0.039 ± 0.016 and the 95% upper limit on r is r < 0.35. The current data still favor the running of ns at 2σ confidence level. 3.3. Dynamical dark energy Assuming the flat universe, first we explore the cosmological constraints in the dark energy model with a constant EoS, w (w ≡ w0 , w1 ≡ 0), from the latest observational data. In Fig. 4 we show the
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Fig. 4. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and w from all datasets together (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters w0 and w1 from all datasets together (red). The blue solid lines stand for w0 = −1 and w0 + w1 = −1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
two-dimensional constraints on w and ns . Current observational data yield a strong constraint on the constant EoS of dark energy, w = −1.060 ± 0.066 (1σ ), which is similar with the limit from WMAP9 [6]. The standard CDM model (w = −1) is consistent with the current observational data. In this case the constraint on ns is slightly changed, ns = 0.960 ± 0.009 at 68% confidence level. For the time evolving EoS, in Fig. 5 we illustrate the constraints on the dark energy parameters w0 and w1 . For the flat universe, due to the limits of the precisions of observational data, the variance of w0 and w1 are still large, namely, the 68% constraints on w0 and w1 are w0 = −1.11 ± 0.15 and w1 = 0.18 ± 0.65. And the 95% constraints are −1.38 < w0 < −0.80 and −1.32 < w1 < 1.15. This result implies that the dynamical dark energy models are not excluded and the current data cannot distinguish different dark energy models decisively. The obtained best fit model is the Quintom dark energymodel [38] with the particular feature that its EoS can cross the cosmological constant boundary smoothly. The standard CDM model, however, is still a good fit right now. For the spectral index, the weak correlations between ns and the parameter of dark energy EoS do not change the limit significantly, namely the 68% constraint is ns = 0.962 ± 0.011. The HZ spectrum is still ruled out at about 4σ confidence level.
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Fig. 6. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and Yp from all datasets together (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.4. Primordial helium abundance A change in the primordial helium abundance affects the shape of the temperature power spectrum [39]. The most dominant effect is a suppression of the power spectrum at > 500, due to an enhanced Silk damping effect. Consequently, in order to match the observational power spectrum on small scales, the spectral index ns need change to compensate. Therefore, ns and Yp are strongly correlated, which is clearly shown in Fig. 6. When we include the primordial helium abundance Yp into analysis, the constraint on ns is significantly weakened from all datasets together, namely ns = 0.976 ± 0.012 at 68% confidence level. The significance of ns < 1 now is at about 2σ confidence level. Meanwhile, we obtain an significant detection of the effect of primordial helium: Yp = 0.298 ± 0.031 (68% C.L.). This value is also consistent with the helium abundances estimated (Yp 0.24−0.25) from observations of low-metallicity extragalactic ionized (HII) regions [40]. 3.5. Neutrino properties We study the constraint on ns in the cosmological models including the neutrino properties, the massive neutrino and the number of relativistic species in this subsection. In Table 2 we show the constraint on the total neutrino mass from all datasets together, namely the 95% upper limit is mν < 0.47 eV, which is consistent with previous works [6,9,11,41,42]. In the massive neutrinos model, the epoch of matter-radiation equality aeq occurs later relative to the standard CDM model, so that at recombination the inhomogeneities induced from the radiation become large. The small-scale anisotropies increases in the massive neutrinos model, which needs a large ns to suppress the power spectrum. In Fig. 7 we show the constraints in the (ns , mν ) panel. Due to the degeneracy, the constraint on ns becomes ns = 0.968 ± 0.009 at 68% confidence level. Then we consider the constraints on the effective number of neutrinos, Neff , from different data combinations (Table 3), assuming massless neutrinos. The effect of Neff on the CMB power spectrum is similar with that of the massive neutrinos. A large value of Neff will lead to a late aeq , which induces an enhanced Silk damping effect. We find the WMAP9 data alone only gives very weak constraint Neff = 4.88 ± 2.08 at the 68% confidence level, which is consistent with the result derived by the WMAP9 team [6]. Adding the small-scale SPT data significantly improves the constraint: Neff = 3.82 ± 0.57 (1σ ). When we combine all datasets together, we
Fig. 7. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and mν from all datasets together (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 3 1σ constraints on some cosmological parameters from different data combinations in the model with the number of relativistic species Neff .
ns Neff H0 100 b h2 100 c h2
CDM + Neff WMAP9 alone WMAP9 + SPT
All datasets
0.994 ± 0.026 4.88 ± 2.08 76.84 ± 8.18 2.259 ± 0.050 14.75 ± 3.79
0.980 ± 0.011 3.79 ± 0.38 72.29 ± 1.82 2.252 ± 0.036 13.04 ± 0.73
0.986 ± 0.018 3.82 ± 0.57 74.31 ± 3.60 2.271 ± 0.048 12.68 ± 1.04
obtain the most stringent constraint on the effective number of neutrinos, namely Neff = 3.79 ± 0.38 (68% C.L.), which is consistent with previous works very well [9,11,23]. This result displays a slight preference for an extra relativistic relic. However, the standard value of Neff = 3.04 remains well within the 95% confidence intervals. Similarly, an enhanced Silk damping effect on small-scale CMB power spectrum produced by a larger Neff can be partially canceled by a larger ns . Neff is strongly correlation with ns . In Fig. 8 we show the constraints on ns and Neff from different data combinations. When using all datasets together, the 68% C.L. constraint on ns becomes ns = 0.980 ± 0.011. The HZ spectrum now is consistent with the current data at 95% confidence level. When we include Neff and the massive neutrinos in the calculations simultaneously, the constraints on parameters become weaker, due to the degeneracies among them. The 95% upper limit of the total neutrino mass is mν < 0.71 eV. The 68% C.L. constraints on Neff and ns becomes Neff = 3.89 ± 0.39 and ns = 0.988 ± 0.013. 3.6. Constraints from Planck Finally, we present the constraints on ns from the first data released by the Planck collaboration. Due to the improved precision, this new Planck data has constrained several cosmological parameters at few percent level. The constraint on the spectral index is significantly improved by the new Planck data. In Table 4 we summarize the constraints on the scalar spectral index in various extended cosmological models from Planck + WP and Planck + WP + BAP + SN data combinations. In the standard CDM model, we obtain the tight constraints on the scalar spectral index of ns = 0.9565 ± 0.0071 and ns = 0.9608 ± 0.0057 at 68% confidence level from Planck + WP and Planck + WP + BAO + SN, respectively. The spectrum with
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4. Summary
Fig. 8. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and Neff from different present data combinations: WMAP9 only (red), WMAP9 + SPT (blue) and all datasets (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 4 1σ constraints on ns from Planck and the combination with BAO and SN. Models/ns
Planck + WP
Planck + WP + BAO + SN
CDM CDM CDM CDM CDM
0.9565 ± 0.0071 0.976 ± 0.013 0.9566 ± 0.0073 0.9604 ± 0.0074 0.9585 ± 0.0074
0.9608 0.9693 0.9613 0.9634 0.9620
+ + + +
Neff mν Yp
αs
± ± ± ± ±
0.0057 0.0079 0.0057 0.0056 0.0057
Recently many experimental groups have published their new observational data, such as temperature and polarization power spectra of WMAP9 [6], temperature power spectrum of SPT at high multipoles [23], and the BAO measurement from the Ly-α forest of BOSS quasars at high redshift z = 2.3 [30]. The WMAP collaboration has presented the cosmological implications of their final nine-year data release, finding that the spectra with the spectral index ns ≥ 1 are disfavored by the current observational data at about 5σ confidence level in the CDM framework. However, in the analyses we find that the strong constraint on ns could be weakened by considering the possible degeneracies between ns and other cosmological parameters introduced in some extended models, such as the tensor fluctuation r and the dark energy EoS w. The largest effect is shown in the model with the number of relativistic species Neff . Due to the strong degeneracy between ns and Neff , the error bar of ns is significantly enlarged, namely ns = 0.980 ± 0.011 (1σ ), and the HZ spectrum now is consistent with the current data at 95% confidence level. Since the Planck collaboration has recently published CMB maps with the highest precision. we also analyze these extended cosmological models again using the Planck data. In the standard CDM, we obtain ns = 0.9608 ± 0.0057 (68% C.L.) from Planck + WP, BAO and Union2.1 compilation data, which has ruled out the HZ scale-invariant spectrum at about 8σ confidence level. However, including Neff could significantly weak the constraint, namely ns = 0.9693 ± 0.0079 (68% C.L.). Now the significance of ns < 1 is only 3σ . Acknowledgements We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. H.L. is supported in part by the National Science Foundation of China under Grant No. 11033005, by the 973 program under Grant No. 2010CB83300, by the Chinese Academy of Science under Grant No. KJCX2-EW-W01. J.X. is supported by the National Youth Thousand Talents Program and the grants No. Y25155E0U1 and No. Y3291740S3. X.Z. is supported in part by the National Science Foundation of China under Grants Nos. 10975142 and 11033005, and by the Chinese Academy of Sciences under Grant No. KJCX3-SYW-N2. References
Fig. 9. Marginalized one-dimensional probability distribution of ns from the data combination of Planck + BAO + SN basing on different cosmological models: the black solid line is given by CDM, the red dash line is given by CDM + α s , the blue dot line is given by CDM + Neff , and the green dash-dot line is given by CDM + Yp . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ns < 1 is strongly favored at very high significance. When we constrain ns in some extended models, the results are only slightly changed relative to the standard CDM model, which is shown in Fig. 9. However, the degeneracy between ns and Neff still strongly affects the constraint on ns . When including Neff into the calculations, the constraints on ns become significantly weaker than those obtained in the standard CDM model, namely 68% C.L. limits are ns = 0.976 ± 0.013 and ns = 0.9693 ± 0.0079 from Planck + WP and Planck + WP + BAO + SN, respectively. The preference of ns < 1 decreases in the CDM + Neff model.
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Contents lists available at ScienceDirect
Physics of the Dark Universe journal homepage: www.elsevier.com/locate/dark
Constraints on scalar spectral index from latest observational measurements夽 Hong Lia , b , * , Jun-Qing Xiaa , Xinmin Zhangc a
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Science, P.O. Box 918-3, Beijing 100049, PR China National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, PR China c Theoretical Physics Division, Institute of High Energy Physics, Chinese Academy of Science, P.O. Box 918-4, Beijing 100049, PR China b
a r t i c l e
i n f o
Keywords: CMB The scalar spectral index
a b s t r a c t With the nine-year data release of the Wilkinson Microwave Anisotropy Probe (WMAP9), it is found that the inflationary models with the scalar spectral index ns ≥ 1 are excluded at about 5σ confidence level. In this paper, we set the new limits on the scalar spectral index in different cosmological models by the WMAP9 data, the small-scale cosmic microwave background (CMB) measurement from the South Pole Telescope, baryon acoustic oscillation data, Hubble Telescope measurements of the Hubble constant, and supernovae luminosity distance data. In most of extended cosmological models, e.g. with a dark energy equation of state, the constraints on ns do not change significantly, when comparing with that obtained in the standard CDM model. The Harrison–Zel’dovich–Peebles (HZ) scale invariant spectrum is still disfavored at more than 4σ confidence level. However, when considering the model with an effective number of neutrinos Neff , we obtain the limit on the spectral index of ns = 0.980 ± 0.011 (1σ ), due to the strong degeneracy between ns and Neff . The HZ spectrum now is consistent with the current data at 95% confidence level. Recently, the Planck collaboration has published CMB maps with the highest precision. Therefore, we also analyze these extended cosmological models again using the Planck data, and find that the degeneracy between ns and Neff still weakens the constraint on the spectral index significantly. c 2013 The Authors. Published by Elsevier B.V. All rights reserved.
1. Introduction Inflation, the most attractive paradigm in the very early universe, has successfully resolved many problems existing in the hot big bang cosmology, such as the flatness, horizon, monopole problem, and so forth [1]. Its quantum fluctuations turn out to be the primordial density fluctuations which seed the observed large scale structures (LSS) and the anisotropies of cosmic microwave background (CMB). To distinguish various inflationary models, the spectral index of the power spectrum of primordial curvature perturbations is one of the most important variables. With the accumulation of observational data from CMB, LSS and Type Ia Supernovae observations (SN) and the improvements of the data quality, the cosmological observations play a crucial role in our understanding of the Universe and also in constraining the cosmological parameters [2–5]. Thus, determining the scalar spectral index
夽 This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike License, which permits non-commercial use, distribution, and reproduction in any medium, provided the original author and source are credited. * Corresponding author at: Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Science, P.O. Box 918-3, Beijing 100049, PR China. Tel.: +86 10 88235124. E-mail address: [email protected] (H. Li).
ns from the observational data is a very powerful and reliable way to understand these inflationary models. The nine-year data release of Wilkinson Microwave Anisotropy Probe (WMAP9) has determined the cosmological parameters accurately and found that the 68% C.L. constraint on the scalar spectral index of ns = 0.9608 ± 0.0080 [6], when combining with the smallscale CMB measurement from the South Pole Telescope (SPT), baryon acoustic oscillation data (BAO) and Hubble Telescope measurements of the Hubble constant (HST). Within the CDM framework, the Harrison–Zel’dovich–Peebles (HZ) scale invariant spectrum (ns ≡ 1) and the spectra with ns > 1 are disfavored at about 5σ confidence level. Recently, the Planck collaboration has released the first cosmological papers providing the highest resolution, full sky, maps of the CMB temperature anisotropies [7]. Due to the improved precision, this new Planck data has constrained several cosmological parameters at few percent level. In the CDM model, the constraint on the spectral index is significantly improved by the new Planck data, namely ns = 0.9603 ± 0.0054 (1σ ) [8]. The spectra with ns ≥ 1 are ruled out at about 8σ confidence level. Although the CDM model is a good candidate for interpreting the data [8], it is still very interesting to investigate constraints on the scalar spectral index in various extended cosmological models, such as the effect number of neutrinos [9,10], the fraction of baryonic mass in primordial helium, the massive neutrino [11,12] or the
c 2013 The Authors. Published by Elsevier B.V. All rights reserved. 2212-6864/$ - see front matter http://dx.doi.org/10.1016/j.dark.2013.11.003
H. Li et al. / Physics of the Dark Universe 2 (2013) 188–194
equation of state of dark energy [13]. More importantly, the degeneracies between ns and these cosmological parameters introduced could weaken the constraints on ns [8,14–18]. In this paper, we explore the cosmological constraints on ns in some extended cosmological models from the latest data sets, including the Planck and WMAP9 power spectra, the small-scale CMB measurement from SPT, the BAO measurements from several LSS surveys, the HST prior on the Hubble constant H0 and the “Union2.1” compilation SN sample made by the Supernova Cosmology Project. Firstly, we consider the general inflationary model with the tensor fluctuations (r) in the CDM framework. We then extend the CDM model allowing for the dark energy models with a constant equation of state (EoS, w) or with a time-varying EoS (w (z)). Finally, we include the massive neutrino case ( mν ) or the effective number of neutrinos (Neff ) into the CDM model. Our paper is organized as follows: In Section 2 we de scribe the method and the latest observational data sets used in the numerical analyses; Section 3 contains our main global constraints of the scalar spectral index ns in different cosmological models from the current observations. The last Section 4 is the conclusions. 2. Method and data 2.1. Numerical method We perform a global fitting of cosmological parameters using the CosmoMC package [19], a Markov Chain Monte Carlo code. We assume purely adiabatic initial conditions and a flat CDM Universe. The following six cosmological parameters are allowed to vary with top-hat priors: the cold dark matter energy density parameter c h2 ∈ [0.01, 0.99], the baryon energy density parameter b h2 ∈ [0.005, 0.1], the scalar spectral index ns ∈ [0.5, 1.5], the primordial amplitude ln [1010 As ] ∈ [2.7, 4.0], the ratio (multiplied by 100) of the sound horizon at decoupling to the angular diameter distance to the last scattering surface 100s ∈ [0.5, 10], and the optical depth to reionization τ ∈ [0.01, 0.8]. The pivot scale is set at ks 0 = 0.05 Mpc−1 . Besides these six basic cosmological parameters, we have several extra cosmological parameters in different extended cosmological models: the running of scalar spectral index α s ≡ d ln ns /d ln k ∈ [−0.1, 0.1]; the tensor to scalar ratio of the primordial spectrum r ≡ At /As ∈ [0, 2]; the fraction of baryonic mass in primordial helium Yp ∈ [0.1, 0.5], the total neutrino mass fraction at the present day
fν ≡
ν h2 mν = ∈ [0, 0.1] ; mh2 93.14 eV mh2
(1)
and the effective number of neutrinos Neff ∈ [0, 10]. We also consider the dark energy model with the EoS parameters w0 ∈ [−2, 0] and w1 ∈ [−5, 2], which is given by the parametrization [20] wde (a) = w0 + w1 (1 − a) ,
(2)
where a ≡ 1/(1 + z) is the scale factor and w1 = −dw/da characterizes the “running” of EoS. The CDM model has w0 = −1 and w1 = 0. For the dark energy model with a constant EoS, w1 = 0. When using the global fitting strategy to constrain the cosmological parameters, it is crucial to include dark energy perturbations [21]. In this paper we use the method provided in Refs. [21,22] to treat the dark energy perturbations consistently in the whole parameter space in the numerical calculations. Therefore, the most general parameter space in the analyses is: b h2 , c h2 , s , τ , ns , As , αs , r, w0 , w1 , Y p , fν , Nef f . (3)
2.2. Current observational data In our analysis, we consider the following cosmological probes: (i) power spectra of CMB temperature and polarization anisotropies;
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(ii) the baryon acoustic oscillation in the galaxy power spectra; (iii) measurement of the current Hubble constant; and (iv) luminosity distances of type Ia supernovae. To incorporate the WMAP9 CMB temperature and polarization power spectra, we use the routines for computing the likelihood supplied by the WMAP team [6]. The WMAP9 polarization data are composed of TE/EE/BB power spectra on large scales (2 ≤ ≤ 23) and TE power spectra on small scales (24 ≤ ≤ 800), while the WMAP9 temperature data includes the CMB anisotropies on scales 2 ≤ ≤ 1200. Furthermore, we also use the recent SPT data [23], using 47 bandpowers in the range 600 ≤ ≤ 3000. The likelihood is assumed to be Gaussian, and we use the published bandpower window functions and covariance matrix. In order to address for foreground contributions, the SZ amplitude, the amplitude of the clustered point source contribution, and the amplitude of the Poisson distributed point source contribution, are added as nuisance parameters in the CMB data analyses. For the Planck data from the 1-year data release [7], we use the low- and high- CMB temperature power spectrum data from Planck with the low- WMAP9 polarization data (Planck + WP). We marginalize over the nuisance parameters that model the unresolved foregrounds with wide priors [24], and do not include the CMB lensing data from Planck [25]. Baryon Acoustic Oscillations provides an efficient method for measuring the expansion history by using features in the clustering of galaxies within large scale surveys as a ruler with which to measure the distance–redshift relation. It provides a particularly robust quantity to measure [26]. It measures not only the angular diameter distance, DA (z), but also the expansion rate of the universe, H(z), which is powerful for studying dark energy [27]. Since the current BAO data are not accurate enough for extracting the information of DA (z) and H(z) separately [28], one can only determine an effective distance [29]: 1/3 D ν (z) = (1 + z)2 D 2A (z) cz/ H (z) .
(4)
In this paper we use the recent BAO measurement at high redshift z = 2.3 detected in the Ly-α forest of Baryon Oscillation Spectroscopic Survey (BOSS) quasars [30]. Furthermore, we also include the BAO measurement from the 6dF Galaxy Redshift Survey (6dFGRS) at a low redshift z = 0.106 [31], and the BAO measurements from the WiggleZ Survey at three redshift bins z = 0.44, z = 0.60 and z = 0.73 [32], the measurement of the BAO scale based on a re-analysis of the Luminous Red Galaxies (LRG) sample from Sloan Digital Sky Survey (SDSS) Data Release 7 at the median redshift z = 0.35 [33], and the BAO signal from BOSS CMASS DR9 data at z = 0.57 [34]. In our analysis, we add a Gaussian prior on the current Hubble constant given by Ref. [35]; H0 = 73.8 ± 2.4 km s−1 Mpc−1 (68% C.L.). The quoted error includes both statistical and systematic errors. This measurement of H0 is obtained from the magnitude–redshift relation of 240 low-z Type Ia supernovae at z < 0.1 by the Near Infrared Camera and Multi-Object Spectrometer (NICMOS) Camera 2 of the Hubble Space Telescope (HST). In addition, we impose a weak top-hat prior on the Hubble parameter: H0 ∈ [40, 100] km s−1 Mpc−1 . Finally, we include data from Type Ia supernovae, which consists of luminosity distance measurements as a function of redshift, DL (z). In this paper we use the latest SN data sets from the Supernova Cosmology Project, “Union Compilation 2.1”, which consists of 580 samples and spans the redshift range 0 ≤ z ≤ 1.55 [36]. This data set also provides the covariance matrix of data with and without systematic errors. In order to be conservative, we use the covariance matrix with systematic errors. When calculating the likelihood from SN, we marginalize over the absolute magnitude M, which is a nuisance parameter, as done in Ref. [37].
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Table 1 1σ constraints on some cosmological parameters from different data combinations in the standard CDM model. Standard CDM WMAP9 alone WMAP9 + SPT ns H0 100 b h2 100 c h2
0.972 70.34 2.270 11.37
± ± ± ±
0.013 2.21 0.050 48
0.966 70.20 2.234 11.41
± ± ± ±
0.011 2.15 0.042 0.47
All datasets 0.963 68.90 2.224 11.70
± ± ± ±
0.008 0.62 0.034 0.17
3. Numerical results In this section we mainly present our global fitting results of the cosmological parameters determined from the latest observational data and focus on the degeneracies between ns and other extended parameters in different models. And finally, we report the constraints on the scalar spectral index from the recent Planck data. 3.1. Standard ΛCDM model Firstly, we consider the standard CDM model. In Table 1 we show the constraints on some related cosmological parameters from three different data combinations: WMAP9 alone, WMAP9 + SPT, and all datasets. In the upper panel of Fig. 1 we show the one-dimensional marginalized likelihood distributions of ns from three data combinations. Using the WMAP9 data alone, we obtain the 68% constraint of ns = 0.972 ± 0.013. The primordial spectra with ns ≥ 1 are only excluded at 2σ confidence level. When we include the smallscale SPT measurement, the constraint on ns becomes slightly tighter, ns = 0.966 ± 0.011 at 1σ confidence level. Since the median value and the error bar of ns are smaller, when comparing with those from WMAP9 alone, the significance of ns < 1 is more than 3σ confidence level. We also show the two-dimensional contour between H0 and ns in the below panel of Fig. 1. As we know, the Hubble constant is anticorrelated with the matter density m . Changing the matter density has effects on the small-scale power spectrum and it needs the spectral index ns to be changed to compensate. Therefore, there is a strong correlation between ns and H0 . When using all datasets together, this degeneracy can be partly broken by the information of H0 measurement. Thus, the constraint on the Hubble constant becomes much more stringent, H0 = 68.90 ± 0.62 km s−1 Mpc−1 (1σ C.L.), due to the HST prior. Consequently, the constraint of the spectral index also becomes tighter significantly, ns = 0.963 ± 0.008 (1σ C.L.). The error bar of ns is reduced by a factor of 1.5, due to the constraining power of BAO, HST and SN. The HZ spectrum is disfavored by the current data at about 5σ confidence level, which is consistent with that from the WMAP9 paper [6]. However, this strong constraint on the spectral index is model dependent apparently. The constraints on ns could be changed, due to the possible degeneracies between ns and other extended parameters in some extended CDM models. In the following subsections, we discuss the constraints on parameters in these extended cosmological models, which is shown in Table 2, such as the inflationary models, α s and r, the dynamical dark energy model, w(z), and ones including the neutrino properties, mν and Neff . 3.2. Inflationary models Firstly, we include the gravitational waves into the analysis. When using all datasets together, the data yield the 95% upper limit of tensor-to-scalar ratio r < 0.15. Meanwhile, the constraint on the spectral index is barely changed, ns = 0.966 ± 0.009 at 68% confidence level, due to the degeneracy between ns and r. In Fig. 2 we show the
Fig. 1. Marginalized one-dimensional and two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and H0 in the standard CDM model from different present data combinations: WMAP9 only (red), WMAP9 + SPT (blue) and all datasets (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 2 1σ constraints on cosmological parameters r, α s , mν , Yp and dark energy EoS from the data combination of WMAP9 + SPT + BAO + SN in different extended models. For the weakly constrained parameters we quote the 95% upper limits instead. Models
Constraints
ns Constraints
CDM + r CDM + α s WCDM W(z)CDM
r < 0.15 α s = −0.023 ± 0.011 w = −1.060 ± 0.066 w0 = −1.11 ± 0.15 w1 = 0.18 ± 0.65 mν < 0.47 eV Yp = 0.298 ± 0.031
0.966 0.948 0.960 0.962 – 0.968 0.976
CDM + mν CDM + Yp
± ± ± ±
0.009 0.011 0.009 0.011
± 0.009 ± 0.012
two-dimensional constraints in the (ns , r) panel which can be compared with the prediction of the inflation models. We find that the HZ scale-invariant spectrum (ns = 1, r = 0) is still disfavored at about 4σ confidence level. Also, the inflation models with “blue” tilt (ns > 1) are excluded by the current observations. Furthermore, assuming the number of e-foldings N = 50−60, the single slow-rolling scalar field with potential V(φ ) ∼ m2 φ 2 , which predicts (ns ,r) = (1 − 2/N, 8/N), is still within the 2σ region, while another single slowrolling scalar field with potential V(φ ) ∼λφ 4 , which predicts (ns ,r) = (1−3/N,16/N), has been excluded more than at 2σ confidence level. We also explore the constraint on the running of the spectral index from the latest observational data. When neglecting the tensor fluctuations (r = 0), the combination of the current observational data yield
H. Li et al. / Physics of the Dark Universe 2 (2013) 188–194
Fig. 2. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and r from all datasets together (red). The two blue solid lines are predicted by the m2 φ 2 and λφ 4 models, respectively. The green points denote predictions assuming that the number of e-foldings N = 50−60 from two models. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and α s from all datasets together with (blue) and without (red) considering the tensor fluctuations in the analysis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the limit on the running of the spectral index of α s = −0.023 ± 0.011 (1σ ), which means the running of ns is favored by the current data at 2σ confidence level. In Fig. 3 we show the two-dimensional constraints in the (ns ,α s ) panel. Due to the degeneracy between ns and α s , the 68% constraint on ns is slightly enlarged, ns = 0.948 ± 0.011. The error bar is relaxed by a factor of 1.5, when comparing with the standard CDM model. Finally, we vary the α s and r simultaneously in the analysis. From the blue contour of Fig. 3, one can see that the constraint on ns does not change, ns = 0.949 ± 0.011 (1σ ). The degeneracy between α s and r significantly weakens the constraints on them, namely the 68% constraint on α s is α s = −0.039 ± 0.016 and the 95% upper limit on r is r < 0.35. The current data still favor the running of ns at 2σ confidence level. 3.3. Dynamical dark energy Assuming the flat universe, first we explore the cosmological constraints in the dark energy model with a constant EoS, w (w ≡ w0 , w1 ≡ 0), from the latest observational data. In Fig. 4 we show the
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Fig. 4. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and w from all datasets together (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters w0 and w1 from all datasets together (red). The blue solid lines stand for w0 = −1 and w0 + w1 = −1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
two-dimensional constraints on w and ns . Current observational data yield a strong constraint on the constant EoS of dark energy, w = −1.060 ± 0.066 (1σ ), which is similar with the limit from WMAP9 [6]. The standard CDM model (w = −1) is consistent with the current observational data. In this case the constraint on ns is slightly changed, ns = 0.960 ± 0.009 at 68% confidence level. For the time evolving EoS, in Fig. 5 we illustrate the constraints on the dark energy parameters w0 and w1 . For the flat universe, due to the limits of the precisions of observational data, the variance of w0 and w1 are still large, namely, the 68% constraints on w0 and w1 are w0 = −1.11 ± 0.15 and w1 = 0.18 ± 0.65. And the 95% constraints are −1.38 < w0 < −0.80 and −1.32 < w1 < 1.15. This result implies that the dynamical dark energy models are not excluded and the current data cannot distinguish different dark energy models decisively. The obtained best fit model is the Quintom dark energymodel [38] with the particular feature that its EoS can cross the cosmological constant boundary smoothly. The standard CDM model, however, is still a good fit right now. For the spectral index, the weak correlations between ns and the parameter of dark energy EoS do not change the limit significantly, namely the 68% constraint is ns = 0.962 ± 0.011. The HZ spectrum is still ruled out at about 4σ confidence level.
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Fig. 6. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and Yp from all datasets together (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.4. Primordial helium abundance A change in the primordial helium abundance affects the shape of the temperature power spectrum [39]. The most dominant effect is a suppression of the power spectrum at > 500, due to an enhanced Silk damping effect. Consequently, in order to match the observational power spectrum on small scales, the spectral index ns need change to compensate. Therefore, ns and Yp are strongly correlated, which is clearly shown in Fig. 6. When we include the primordial helium abundance Yp into analysis, the constraint on ns is significantly weakened from all datasets together, namely ns = 0.976 ± 0.012 at 68% confidence level. The significance of ns < 1 now is at about 2σ confidence level. Meanwhile, we obtain an significant detection of the effect of primordial helium: Yp = 0.298 ± 0.031 (68% C.L.). This value is also consistent with the helium abundances estimated (Yp 0.24−0.25) from observations of low-metallicity extragalactic ionized (HII) regions [40]. 3.5. Neutrino properties We study the constraint on ns in the cosmological models including the neutrino properties, the massive neutrino and the number of relativistic species in this subsection. In Table 2 we show the constraint on the total neutrino mass from all datasets together, namely the 95% upper limit is mν < 0.47 eV, which is consistent with previous works [6,9,11,41,42]. In the massive neutrinos model, the epoch of matter-radiation equality aeq occurs later relative to the standard CDM model, so that at recombination the inhomogeneities induced from the radiation become large. The small-scale anisotropies increases in the massive neutrinos model, which needs a large ns to suppress the power spectrum. In Fig. 7 we show the constraints in the (ns , mν ) panel. Due to the degeneracy, the constraint on ns becomes ns = 0.968 ± 0.009 at 68% confidence level. Then we consider the constraints on the effective number of neutrinos, Neff , from different data combinations (Table 3), assuming massless neutrinos. The effect of Neff on the CMB power spectrum is similar with that of the massive neutrinos. A large value of Neff will lead to a late aeq , which induces an enhanced Silk damping effect. We find the WMAP9 data alone only gives very weak constraint Neff = 4.88 ± 2.08 at the 68% confidence level, which is consistent with the result derived by the WMAP9 team [6]. Adding the small-scale SPT data significantly improves the constraint: Neff = 3.82 ± 0.57 (1σ ). When we combine all datasets together, we
Fig. 7. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and mν from all datasets together (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 3 1σ constraints on some cosmological parameters from different data combinations in the model with the number of relativistic species Neff .
ns Neff H0 100 b h2 100 c h2
CDM + Neff WMAP9 alone WMAP9 + SPT
All datasets
0.994 ± 0.026 4.88 ± 2.08 76.84 ± 8.18 2.259 ± 0.050 14.75 ± 3.79
0.980 ± 0.011 3.79 ± 0.38 72.29 ± 1.82 2.252 ± 0.036 13.04 ± 0.73
0.986 ± 0.018 3.82 ± 0.57 74.31 ± 3.60 2.271 ± 0.048 12.68 ± 1.04
obtain the most stringent constraint on the effective number of neutrinos, namely Neff = 3.79 ± 0.38 (68% C.L.), which is consistent with previous works very well [9,11,23]. This result displays a slight preference for an extra relativistic relic. However, the standard value of Neff = 3.04 remains well within the 95% confidence intervals. Similarly, an enhanced Silk damping effect on small-scale CMB power spectrum produced by a larger Neff can be partially canceled by a larger ns . Neff is strongly correlation with ns . In Fig. 8 we show the constraints on ns and Neff from different data combinations. When using all datasets together, the 68% C.L. constraint on ns becomes ns = 0.980 ± 0.011. The HZ spectrum now is consistent with the current data at 95% confidence level. When we include Neff and the massive neutrinos in the calculations simultaneously, the constraints on parameters become weaker, due to the degeneracies among them. The 95% upper limit of the total neutrino mass is mν < 0.71 eV. The 68% C.L. constraints on Neff and ns becomes Neff = 3.89 ± 0.39 and ns = 0.988 ± 0.013. 3.6. Constraints from Planck Finally, we present the constraints on ns from the first data released by the Planck collaboration. Due to the improved precision, this new Planck data has constrained several cosmological parameters at few percent level. The constraint on the spectral index is significantly improved by the new Planck data. In Table 4 we summarize the constraints on the scalar spectral index in various extended cosmological models from Planck + WP and Planck + WP + BAP + SN data combinations. In the standard CDM model, we obtain the tight constraints on the scalar spectral index of ns = 0.9565 ± 0.0071 and ns = 0.9608 ± 0.0057 at 68% confidence level from Planck + WP and Planck + WP + BAO + SN, respectively. The spectrum with
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4. Summary
Fig. 8. Marginalized two-dimensional likelihood (1, 2σ contours) constraints on the parameters ns and Neff from different present data combinations: WMAP9 only (red), WMAP9 + SPT (blue) and all datasets (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 4 1σ constraints on ns from Planck and the combination with BAO and SN. Models/ns
Planck + WP
Planck + WP + BAO + SN
CDM CDM CDM CDM CDM
0.9565 ± 0.0071 0.976 ± 0.013 0.9566 ± 0.0073 0.9604 ± 0.0074 0.9585 ± 0.0074
0.9608 0.9693 0.9613 0.9634 0.9620
+ + + +
Neff mν Yp
αs
± ± ± ± ±
0.0057 0.0079 0.0057 0.0056 0.0057
Recently many experimental groups have published their new observational data, such as temperature and polarization power spectra of WMAP9 [6], temperature power spectrum of SPT at high multipoles [23], and the BAO measurement from the Ly-α forest of BOSS quasars at high redshift z = 2.3 [30]. The WMAP collaboration has presented the cosmological implications of their final nine-year data release, finding that the spectra with the spectral index ns ≥ 1 are disfavored by the current observational data at about 5σ confidence level in the CDM framework. However, in the analyses we find that the strong constraint on ns could be weakened by considering the possible degeneracies between ns and other cosmological parameters introduced in some extended models, such as the tensor fluctuation r and the dark energy EoS w. The largest effect is shown in the model with the number of relativistic species Neff . Due to the strong degeneracy between ns and Neff , the error bar of ns is significantly enlarged, namely ns = 0.980 ± 0.011 (1σ ), and the HZ spectrum now is consistent with the current data at 95% confidence level. Since the Planck collaboration has recently published CMB maps with the highest precision. we also analyze these extended cosmological models again using the Planck data. In the standard CDM, we obtain ns = 0.9608 ± 0.0057 (68% C.L.) from Planck + WP, BAO and Union2.1 compilation data, which has ruled out the HZ scale-invariant spectrum at about 8σ confidence level. However, including Neff could significantly weak the constraint, namely ns = 0.9693 ± 0.0079 (68% C.L.). Now the significance of ns < 1 is only 3σ . Acknowledgements We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. H.L. is supported in part by the National Science Foundation of China under Grant No. 11033005, by the 973 program under Grant No. 2010CB83300, by the Chinese Academy of Science under Grant No. KJCX2-EW-W01. J.X. is supported by the National Youth Thousand Talents Program and the grants No. Y25155E0U1 and No. Y3291740S3. X.Z. is supported in part by the National Science Foundation of China under Grants Nos. 10975142 and 11033005, and by the Chinese Academy of Sciences under Grant No. KJCX3-SYW-N2. References
Fig. 9. Marginalized one-dimensional probability distribution of ns from the data combination of Planck + BAO + SN basing on different cosmological models: the black solid line is given by CDM, the red dash line is given by CDM + α s , the blue dot line is given by CDM + Neff , and the green dash-dot line is given by CDM + Yp . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ns < 1 is strongly favored at very high significance. When we constrain ns in some extended models, the results are only slightly changed relative to the standard CDM model, which is shown in Fig. 9. However, the degeneracy between ns and Neff still strongly affects the constraint on ns . When including Neff into the calculations, the constraints on ns become significantly weaker than those obtained in the standard CDM model, namely 68% C.L. limits are ns = 0.976 ± 0.013 and ns = 0.9693 ± 0.0079 from Planck + WP and Planck + WP + BAO + SN, respectively. The preference of ns < 1 decreases in the CDM + Neff model.
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